Clone of What numbers can we make?

8279 /www/nrich/html/content/id/8279/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span style="font-weight: bold;">Imagine you had four bags containing a large number of</span> <span style="font-weight: bold;">1s, 4s, 7s and 10s.</span></p> <p>You can choose numbers from the bags and add them to make different totals. You don't have to use numbers from every bag, and there will always be as many of each number as you need.</p> <p> </p> <div style="text-align: center;"><mdo:image src="14710bag.png"></mdo:image></div> <p> </p> <p>Choose some sets of $3$ numbers and add them together.<br></br> <strong>What is special about your answers?<br></br> Can you explain what you've noticed?</strong></p> <p><br></br> Charlie and Alison came up with some ways to represent what was happening.</p> <p><strong>Charlie's representation:</strong><br></br> </p> <div class="toggle"> <p style="margin-left: 40px;">All multiples of three can be represented as:</p> <p style="text-align: center; margin-left: 40px;"><mdo:image src="Charlie3n.png"></mdo:image></p> <p style="margin-left: 40px;">The numbers in the bags can be represented as:</p> <p style="text-align: center; margin-left: 40px;"><mdo:image src="Charlie3n%2B1.png"></mdo:image></p> <p style="margin-left: 40px;"><em>Similarly, numbers which are two more than a multiple of three can be represented as:</em></p> <p style="text-align: center; margin-left: 40px;"><mdo:image src="Charlie3n%2B2.png"></mdo:image></p> <p style="margin-left: 40px;">When I choose three numbers, I end up with a multiple of three $+3$ which will be a multiple of three.</p> <p style="margin-left: 40px;"><mdo:image src="Charliesum.png"></mdo:image></p> <p> </p> </div> <p><br></br> <strong>Alison's representation:</strong><br></br> </p> <div class="toggle"> <p style="margin-left: 40px;">Since all multiples of three can be written in the form $3n$, the numbers in the bags can be written in the form $3n+1$.</p> <p style="margin-left: 40px;"><em>Similarly, numbers which are two more than a multiple of three can be written in the form $3n+2$.</em></p> <p style="margin-left: 40px;">As long as I remember I'm working with multiples of three, I could call these numbers $+0$, $+1$ and $+2$ numbers for short.</p> <p style="margin-left: 40px;">When I choose three numbers, I'm adding together three $+1$s, so I end up with a multiple of three $+3$ which will be a multiple of three.</p> <p> </p> </div> <p> </p> <p>What if you choose sets of $4$ numbers and add them together?<br></br> What if you choose sets of $5$ numbers, $6$ numbers, $7$ numbers...?<br></br> What totals do you think it would be possible to make if you choose $99$ numbers? Or 100 numbers?</p> <p><strong>Can you use Charlie's and Alison's representations to convince yourself?</strong></p> <p> </p> <p>Next, you might like to work on <a href="/1866">Take Three From Five</a>,<br></br> or <a href="/8280">What Numbers Can We Make Now?</a></p> <p><br></br> <br></br> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">We received some well considered solutions to this problem.</span> <br></br> <br></br> <span class="editorial">Tom and Freya from Ide Primary School found five ways of making 16 using four numbers:</span><br></br> <br></br> 7+7+1+1=16<br></br> 5+5+3+3=16<br></br> 7+3+3+3=16<br></br> 5+7+3+1=16<br></br> 5+5+5+1=16 <br></br> <br></br> <br></br> <span class="editorial">Daniel from Staplehurst School found that it was impossible to make 25 with six odd numbers and then went on to list the possible totals that could be made from selecting three, four, five or six numbers from the set of bags below.</span> <a class="editorial" href="/content/id/7405/Daniel.pdf">Here</a> <span class="editorial">is a summary of his findings.</span><br></br> <br></br> <br></br> <div style="text-align: center;"> <mdo:image width="417" height="138" src="14710bag.jpg" alt=""></mdo:image></div> <div style="text-align: left;"> </div> <div style="text-align: left;"><span class="editorial">Brandyn from Garden International School started by exploring this set of bags and then considered other sets of bags:</span></div> <div style="text-align: left;"> </div> <div style="text-align: left;">Choosing just three numbers from the bags above gave the following totals, all multiples of 3:</div> <br></br> <table border="1"> <tbody> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">4</td> <td style="text-align: center;">7</td> <td style="text-align: center;">10</td> <td style="text-align: center;">TOTAL</td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">12</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;">21</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;">30</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">6</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">9</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">12</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">9</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">15</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">18</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">15</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">18</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">24</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">21</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">24</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">27</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">12</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">15</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">18</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">21</td> </tr> </tbody> </table> <br></br> <br></br> Choosing four numbers from the bags above gave the following totals, all 1 more than (or 2 less than) multiples of 3:<br></br> <br></br> <table border="1"> <tbody> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">4</td> <td style="text-align: center;">7</td> <td style="text-align: center;">10</td> <td style="text-align: center;">TOTAL</td> </tr> <tr> <td style="text-align: center;">4</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">4</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">4</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">16</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">4</td> <td style="text-align: center;"> </td> <td style="text-align: center;">28</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">4</td> <td style="text-align: center;">40</td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">7</td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">10</td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">13</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">13</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">19</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">22</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;">22</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;">25</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;">1</td> <td style="text-align: center;">31</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;">31</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">3</td> <td style="text-align: center;">34</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">3</td> <td style="text-align: center;">37</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">10</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">16</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">22</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">22</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">28</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">2</td> <td style="text-align: center;">34</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">13</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">16</td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">19</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">16</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">19</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">25</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;">19</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">25</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">1</td> <td style="text-align: center;">28</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">2</td> <td style="text-align: center;">25</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">28</td> </tr> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">2</td> <td style="text-align: center;">31</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">1</td> <td style="text-align: center;">22</td> </tr> </tbody> </table> <div> </div> <div>If I choose 5 numbers I predict that the series will start with 5 and increase in 3's.</div> <div> </div> <div>If I choose 6 numbers I predict that the series will start with 6 and increase in 3's, etc..</div> <div> </div> <div>If I choose 99 numbers, I predict that the series will start with 99 and increase in 3's.</div> <div> </div> <div style="font-weight: bold;">Bags containing 2's, 5's, 8's and 11's</div> <div> </div> <div>If the bags had contained 2's, 5's, 8's and 11s, the series would start with the amount of numbers you are choosing (e.g. 4) multiplied by 2.</div> <div> </div> <div>They would start with 8 first because 4 x 2 is 8 and then increase by 3.</div> <div> </div> <div style="font-weight: bold;">Bags containing 2's, 7's, 12's and 17's</div> <div> </div> <div>I predict that the number sequence will start with the number of numbers multiplied by 2 and then increasing by 5 each time, as the number that it increases is the difference between two adjacent bags.</div> <div> </div> <div>An example would be if I choose 100 numbers, the series would start with 200 and then go to 205, 210, 215, etc..</div> <div> </div> <div style="font-weight: bold;">The formula is:</div> <div style="font-weight: bold;"> </div> <div>The number of numbers multiplied by the number in the first bag is the starting number.</div> <div>The sequence then increases by the difference between two adjacent bags.</div> <div> </div> <div style="font-weight: bold;">Bags containing 3's, 7's, 11's and 15's</div> <div> </div> <div>To find out if 412 can be made by choosing 30 numbers with bags of 3's, 7's, 11's and 15's, I would do the following:</div> <div> </div> <div>I would multiply 30 by 3 to give me the starting number (90). Then I would subtract 90 from 412 which gives me an answer of 322. Next I would divide 322 by 4 (the difference between adjacent bags) and that answer will be 80.5</div> <div>Since this is not a whole number, we cannot get 412 by choosing 30 numbers. </div> <br></br> <br></br> <span class="editorial">Krystof from Uhelny Trh School in Prague investigated what numbers could be made when you choose 3, 4, 5, 6, ... 20, ... 99, 100 numbers, from various sets of bags.</span><br></br> <a class="editorial" href="/content/id/7405/Krystof.pdf">Here</a> <span class="editorial">is the summary of his findings.</span><br></br> <br></br> <span class="editorial">Well done to you all.</span> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> <p>This problem offers students the opportunity to consider the underlying structure behind multiples and remainders, as well as leading to some very nice generalisations and justifications.<br></br> </p> <h3>Possible approach</h3> <p>Display the image of the four bags (available as a PowerPoint slide here).</p> <p> </p> <div>"Here are some bags containing lots of $1$s, $4$s, $7$s and $10$s. I'd like you to choose any three numbers from the bags and add them together."</div> <div>Clarify that they can reuse numbers if they like, then collect some of their answers on the board.</div> <div> </div> <div>"What do you notice?" <em>"They're all multiples of three".</em></div> <div>"Try to find three numbers from the bags that don't add up to a multiple of three. If you can't do it, see if you can come up with an explanation of why it's not possible."</div> <div> </div> <div>Give students some time to work in pairs. While they are working, circulate and listen to their ideas. Then bring the class together to share insights.</div> <div> </div> <div>If the class have not developed a useful representation of their own, you could introduce Charlie's and/or Alison's representation from the problem.</div> <div> </div> <div>"Next, I'd like you to explore what happens if you select four numbers from the bag, or five, or six, or... In a while I'm going to pick a large number at random. Your challenge will be to tell me what is special about the total if I selected that many numbers from the set of bags."</div> <div> </div> <div>Give students time to work in pairs on the task. Then bring the class together and choose a large number such as $99$ or $100$: "What would be special about the total if I added $99$ numbers chosen from the bags?" Make sure students explain their reasoning with reference to the structure of the problem rather than just by spotting a pattern.</div> <div> </div> <br></br> <h3>Key questions</h3> <div>What's the smallest number I can make?</div> <div>What's the next smallest?</div> <div>What is special about the numbers that make up each set of bags?</div> <h3> </h3> <h3>Possible extension</h3> <p><a href="http://nrich.maths.org/1866&part=">Take Three from Five</a> and <a href="/8280">What Numbers Can We Make Now?</a> are suitable follow-up problems.</p> <br></br> <h3>Possible support</h3> <p>Begin by asking students to explore what happens when they add two, three, four... numbers chosen from a set of bags containing $2$s, $4$s, $6$s and $8$s. Can they explain their findings?<br></br> <br></br> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p>Begin by exploring what happens when you add two, three, four... numbers chosen from a set of bags containing $2$s, $4$s, $6$s and $8$s. Can you explain your findings?</p></mdoxml> 2 4 0 0 1 0 0 Clone of What numbers can we make? Imagine you had four bags containing a large number of 1s, 3s, 5s and 7s. Can you find ways of choosing six numbers whose total is 25? Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof